Let S be the sum of the first 9 terms of the series :
`{x+ka}+{x^(2)+(k+2)a}+{x^(3)+(k+4)a}+{x^(4)+(k+6)a}+...` where `a!=0` and `a!=1`.
If `S=(x^(10)-x+45a(x-1))/(x-1)`, then k is equal to :

Options

A. 3
B. -3
C. 1
D. -5

Answer

To solve the problem, we need to find the value of \( k \) given the sum \( S \) of the first 9 terms of the series:

\[ S = \frac{x^{10} - x + 45a(x-1)}{x-1} \]

The series is defined as:

\[ \{x + ka\}, \{x^2 + (k + 2)a\}, \{x^3 + (k + 4)a\}, \{x^4 + (k + 6)a\}, \ldots \]

### Step 1: Identify the general term of the series

The \( r \)-th term of the series can be expressed as:

\[ T_r = x^r + (k + 2(r - 1))a \]

This simplifies to:

\[ T_r = x^r + (k + 2r - 2)a \]

### Step 2: Write the sum of the first 9 terms

The sum \( S \) of the first 9 terms is:

\[ S = \sum_{r=1}^{9} T_r = \sum_{r=1}^{9} \left( x^r + (k + 2r - 2)a \right) \]

This can be separated into two sums:

\[ S = \sum_{r=1}^{9} x^r + \sum_{r=1}^{9} (k + 2r - 2)a \]

### Step 3: Calculate the first sum

The first sum is a geometric series:

\[ \sum_{r=1}^{9} x^r = x + x^2 + x^3 + \ldots + x^9 = \frac{x(x^9 - 1)}{x - 1} \]

### Step 4: Calculate the second sum

The second sum can be simplified as follows:

\[ \sum_{r=1}^{9} (k + 2r - 2)a = 9ka + 2a\sum_{r=1}^{9} r - 2 \cdot 9a \]

Using the formula for the sum of the first \( n \) natural numbers:

\[ \sum_{r=1}^{9} r = \frac{9 \cdot 10}{2} = 45 \]

Thus, we have:

\[ \sum_{r=1}^{9} (k + 2r - 2)a = 9ka + 2a(45) - 18a = 9ka + 90a - 18a = 9ka + 72a \]

### Step 5: Combine the sums

Now we can combine both sums:

\[ S = \frac{x(x^9 - 1)}{x - 1} + (9ka + 72a) \]

### Step 6: Set the expression for \( S \)

We know from the problem statement that:

\[ S = \frac{x^{10} - x + 45a(x - 1)}{x - 1} \]

### Step 7: Equate the two expressions for \( S \)

Equating the two expressions for \( S \):

\[ \frac{x(x^9 - 1)}{x - 1} + (9ka + 72a) = \frac{x^{10} - x + 45a(x - 1)}{x - 1} \]

### Step 8: Clear the denominators

Multiply through by \( x - 1 \):

\[ x(x^9 - 1) + (9ka + 72a)(x - 1) = x^{10} - x + 45a(x - 1) \]

### Step 9: Expand and simplify

Expanding both sides gives:

\[ x^{10} - x + 9kax - 9ka + 72ax - 72a = x^{10} - x + 45ax - 45a \]

### Step 10: Collect like terms

Collecting like terms, we can set the coefficients equal to each other:

1. Coefficient of \( x \): \( 9ka + 72a = 45a \)
2. Constant terms: \( -9ka - 72a = -45a \)

From the first equation:

\[ 9ka + 72a = 45a \implies 9ka = 45a - 72a \implies 9ka = -27a \implies k = -3 \]

### Final Answer

Thus, the value of \( k \) is:

\[ \boxed{-3} \]